210:, previously used to show that the clique number required exponentially large monotone circuits, also shows that the Tardos function requires exponentially large monotone circuits despite being computable by a non-monotone circuit of polynomial size. Later, the same function was used to provide a
93:. Approximating the Lovász number of the complement and then rounding the approximation to an integer would not necessarily produce a monotone function, however. To make the result monotone, Tardos approximates the Lovász number of the complement to within an additive error of
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The Tardos function is monotone, in the sense that adding edges to a graph can only cause its Tardos function to increase or stay the same, but never decrease.
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Tardos used her function to prove an exponential separation between the capabilities of monotone
Boolean logic circuits and arbitrary circuits. A result of
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321:(1981), "The ellipsoid method and its consequences in combinatorial optimization",
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390:; Boppana, R. B. (1987), "The monotone circuit complexity of Boolean functions",
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to the approximation, and then rounds the result to the nearest integer. Here
361:(1985), "Lower bounds on the monotone complexity of some Boolean functions",
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243:"The gap between monotone and nonmonotone circuit complexity is exponential"
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290:, Algorithms and Combinatorics, vol. 27, Springer, p. 272,
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for computing the Tardos function requires exponential size.
441:"On Norbert Blum's claimed proof that P does not equal NP"
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denotes the number of edges in the given graph, and
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Boolean
Function Complexity: Advances and Frontiers
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56:of the graph. These two numbers are both
91:Grötschel, Lovász & Schrijver (1981)
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66:The Tardos function can be computed in
81:To define her function, Tardos uses a
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85:for the Lovász number, based on the
83:polynomial-time approximation scheme
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203:denotes the number of vertices.
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363:Doklady Akademii Nauk SSSR
214:to a purported proof of
156:{\displaystyle m/n^{2}}
121:{\displaystyle 1/n^{2}}
284:Jukna, Stasys (2012),
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439:(August 15, 2017),
469:Circuit complexity
416:10.1007/BF02579196
337:10.1007/BF02579273
264:10.1007/BF02122563
208:Alexander Razborov
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22:circuit complexity
218:by Norbert Blum.
196:{\displaystyle n}
176:{\displaystyle m}
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393:Combinatorica
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324:Combinatorica
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319:Schrijver, A.
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311:Grötschel, M.
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297:9783642245084
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18:graph theory
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400:(1): 1–22,
60:to compute.
458:Categories
315:Lovász, L.
239:Tardos, É.
222:References
34:Éva Tardos
445:in theory
402:CiteSeerX
40:Like the
388:Alon, N.
241:(1988),
52:and the
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375:0785629
345:0625550
272:0952004
128:, adds
58:NP-hard
44:of the
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216:P ≠NP
24:, the
246:(PDF)
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329:1
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256:8
191:n
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101:1
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